Question

Fatima was asked to use and identify appropriate properties of equality to find the value of y that makes the statement ¾ (y + 8) = ½ (2y + 4) true. She has written out her steps below. Label those steps with the property used.
4 (¾ (y + 8)) = 4(¼ (2y + 4)) (_____________________)
3 (y + 8) = 2y + 4 (simplify)
3(y) + 3(8) = 2y + 4 (_____________________)
3y + 24 = 2y + 4 (simplify)
3y – 2y + 24 = 2y – 2y + 4 (_____________________)
y + 24 = 4 (combine like terms)
y + 24 – 24 = 4 – 24 (_____________________)
y = -20

(show your work and like a 9th grader wrote it)

Answer 1

Sure! Let's go through Fatima's steps and label them with the appropriate properties of equality:

1. **4 (¾ (y + 8)) = 4 (½ (2y + 4))** (Multiplication Property of Equality)
- This step shows that she multiplied both sides of the equation by 4 to eliminate the fractions.

2. **3 (y + 8) = 2y + 4** (Simplify)
- Here, she simplified the expressions on both sides.

3. **3(y) + 3(8) = 2y + 4** (Distributive Property)
- In this step, she applied the distributive property to distribute 3 across the terms in the parentheses.

4. **3y + 24 = 2y + 4** (Simplify)
- This is the simplification step, combining the products from the previous step.

5. **3y – 2y + 24 = 2y – 2y + 4** (Subtraction Property of Equality)
- In this step, she subtracted \(2y\) from both sides, maintaining equality.

6. **y + 24 = 4** (Combine like terms)
- At this point, she combined like terms to simplify the equation.

7. **y + 24 – 24 = 4 – 24** (Subtraction Property of Equality)
- Here, she subtracted 24 from both sides to isolate \(y\).

8. **y = -20** (Final Result)
- This is the solution for \(y\).

So, the labeled steps would look like this:

1. **4 (¾ (y + 8)) = 4 (½ (2y + 4))** (Multiplication Property of Equality)
2. **3 (y + 8) = 2y + 4** (Simplify)
3. **3(y) + 3(8) = 2y + 4** (Distributive Property)
4. **3y + 24 = 2y + 4** (Simplify)
5. **3y – 2y + 24 = 2y – 2y + 4** (Subtraction Property of Equality)
6. **y + 24 = 4** (Combine like terms)
7. **y + 24 – 24 = 4 – 24** (Subtraction Property of Equality)
8. **y = -20** (Final Result)

Fatima correctly used properties of equality to solve for \(y\)!